Heat oil in a large saucepan over medium heat. Add shallot and cook, stirring, until beginning to soften, about 1 minute. Add port and figs and cook, stirring, 1 minute. Add 1 1/2 cups strawberries, 1 cup broth, 1 tablespoon vinegar, 1/4 teaspoon salt and 1/4 teaspoon pepper; bring to a boil. Reduce heat to a simmer and cook, stirring occasionally, until the strawberries are very soft and broken down and the sauce has reduced slightly, about 15 minutes.

Strain the sauce through a fine-mesh strainer into a 2-cup glass measuring cup or bowl (discard the solids). You should have about 1 cup liquid. (If you have more than that, return the sauce to the pan and continue reducing until you have about 1 cup.) Combine the remaining 2 tablespoons broth and cornstarch in a small bowl. Return the strained sauce to the pan along with the cornstarch mixture; cook over medium heat, stirring, until the sauce is thick enough to coat the back of a spoon and has reduced by half, 5 to 7 minutes. Combine 1 tablespoon of the sauce with the remaining 1 teaspoon vinegar in a small bowl to use as a basting sauce. Cover the remaining sauce to keep warm.

Meanwhile, preheat grill to medium.

Season duck breasts with the remaining 1/4 teaspoon salt and pepper. Oil the grill rack (see Tip). Grill the duck, basting twice, until an instant-read thermometer inserted into the thickest part registers 150°F for medium, 4 to 8 minutes per side, depending on the thickness. Transfer to a clean cutting board and let rest for 5 minutes.

While the duck is resting, chop the remaining 1 cup strawberries. Slice the duck and serve with the sauce, garnished with the chopped strawberries and basil.